1. Consider the function of two variables defined by p f (x, y) = 1 x2 + y 2 , x R, y R . (a) Compute the level sets f (x, y) = c for levels c = 0, 12 , 1, and use these to construct a sketch of z = f (x, y). Describe the surface and mention or label its important features. (b) Find the partial derivatives f (x, y) and x f (x, y) . y (c) Find equations of the planes tangent to the graph of f at the points (1, 1) and \u0010 \u0011 1 1 2 , 2 . (d) Calculate the limits f (x, y) and (x,y)(0,0) x lim f (x, y) (x,y)(0,0) y lim or prove that either or both do not exist. Thus determine the equation of the tangent plane to z = f (x, y) at (0, 0), or explain why it does not exist. 2. On the domain \b D = (x, y) R2 | x [0, 2], y [0, 2] consider the function f (x, y) = 2x3 + 3y 2 6xy + 2 . (a) Sketch the domain D on the x-y plane. (b) Find any critical points of the function f (x, y) inside the domain D that are not on the boundary of D. (c) Find the global maximum and global minimum of f (x, y) on the domain D. 2